Now we stipulate that the value of the sum of two games should be the sum of those games' values. (To express this symbolically: Let x+S, where x is a number and S is a set of numbers, denote the set obtained by adding x to all elements of S. Then we can define addition recursively: where x = {xL | xR} and y = {yL | yR}, x+y = {y+xL U x+yL | y+xR U x+yR}.) In particular, if playing two games simul